Linear Algebra Examples

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}]$

Step 1

Find the eigenvalues.

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Step 1.1

Set up the formula to find the characteristic equation $p (λ)$ .

$p (λ) = determinant (A - λ I_{3})$

Step 1.2

The identity matrix or unit matrix of size $3$ is the $3 \times 3$ square matrix with ones on the main diagonal and zeros elsewhere.

$[\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Step 1.3

Substitute the known values into $p (λ) = determinant (A - λ I_{3})$ .

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Step 1.3.1

Substitute $[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}]$ for $A$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] - λ I_{3})$

Step 1.3.2

Substitute $[\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$ for $I_{3}$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] - λ [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}])$

Step 1.4

Simplify.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Multiply $- λ$ by each element of the matrix.

$p (λ) = determinant ([\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2

Simplify each element in the matrix.

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Step 1.4.1.2.1

Multiply $- 1$ by $1$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} - λ & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.2

Multiply $- λ \cdot 0$ .

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Step 1.4.1.2.2.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} - λ & 0 λ & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.2.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.3

Multiply $- λ \cdot 0$ .

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Step 1.4.1.2.3.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 λ \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.3.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.4

Multiply $- λ \cdot 0$ .

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Step 1.4.1.2.4.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 λ & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.4.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.5

Multiply $- 1$ by $1$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.6

Multiply $- λ \cdot 0$ .

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Step 1.4.1.2.6.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 λ \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.6.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.7

Multiply $- λ \cdot 0$ .

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Step 1.4.1.2.7.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 λ & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.7.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.8

Multiply $- λ \cdot 0$ .

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Step 1.4.1.2.8.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 λ & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.8.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.9

Multiply $- 1$ by $1$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 & - λ \end{array}])$

Step 1.4.2

Add the corresponding elements.

$p (λ) = determinant [\begin{array}{c} 1 - λ & - 4 + 0 & 2 + 0 \\ 2 + 0 & - 7 - λ & 0 + 0 \\ 3 + 0 & - 11 + 0 & 2 - λ \end{array}]$

Step 1.4.3

Simplify each element.

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Step 1.4.3.1

Add $- 4$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 1 - λ & - 4 & 2 + 0 \\ 2 + 0 & - 7 - λ & 0 + 0 \\ 3 + 0 & - 11 + 0 & 2 - λ \end{array}]$

Step 1.4.3.2

Add $2$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 1 - λ & - 4 & 2 \\ 2 + 0 & - 7 - λ & 0 + 0 \\ 3 + 0 & - 11 + 0 & 2 - λ \end{array}]$

Step 1.4.3.3

Add $2$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 1 - λ & - 4 & 2 \\ 2 & - 7 - λ & 0 + 0 \\ 3 + 0 & - 11 + 0 & 2 - λ \end{array}]$

Step 1.4.3.4

Add $0$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 1 - λ & - 4 & 2 \\ 2 & - 7 - λ & 0 \\ 3 + 0 & - 11 + 0 & 2 - λ \end{array}]$

Step 1.4.3.5

Add $3$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 1 - λ & - 4 & 2 \\ 2 & - 7 - λ & 0 \\ 3 & - 11 + 0 & 2 - λ \end{array}]$

Step 1.4.3.6

Add $- 11$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 1 - λ & - 4 & 2 \\ 2 & - 7 - λ & 0 \\ 3 & - 11 & 2 - λ \end{array}]$

Step 1.5

Find the determinant.

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Step 1.5.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $2$ by its cofactor and add.

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Step 1.5.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 1.5.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 1.5.1.3

The minor for $a_{21}$ is the determinant with row $2$ and column $1$ deleted.

$| \begin{array}{c} - 4 & 2 \\ - 11 & 2 - λ \end{array} |$

Step 1.5.1.4

Multiply element $a_{21}$ by its cofactor.

$- 2 | \begin{array}{c} - 4 & 2 \\ - 11 & 2 - λ \end{array} |$

Step 1.5.1.5

The minor for $a_{22}$ is the determinant with row $2$ and column $2$ deleted.

$| \begin{array}{c} 1 - λ & 2 \\ 3 & 2 - λ \end{array} |$

Step 1.5.1.6

Multiply element $a_{22}$ by its cofactor.

$(- 7 - λ) | \begin{array}{c} 1 - λ & 2 \\ 3 & 2 - λ \end{array} |$

Step 1.5.1.7

The minor for $a_{23}$ is the determinant with row $2$ and column $3$ deleted.

$| \begin{array}{c} 1 - λ & - 4 \\ 3 & - 11 \end{array} |$

Step 1.5.1.8

Multiply element $a_{23}$ by its cofactor.

$0 | \begin{array}{c} 1 - λ & - 4 \\ 3 & - 11 \end{array} |$

Step 1.5.1.9

Add the terms together.

$p (λ) = - 2 | \begin{array}{c} - 4 & 2 \\ - 11 & 2 - λ \end{array} | + (- 7 - λ) | \begin{array}{c} 1 - λ & 2 \\ 3 & 2 - λ \end{array} | + 0 | \begin{array}{c} 1 - λ & - 4 \\ 3 & - 11 \end{array} |$

Step 1.5.2

Multiply $0$ by $| \begin{array}{c} 1 - λ & - 4 \\ 3 & - 11 \end{array} |$ .

$p (λ) = - 2 | \begin{array}{c} - 4 & 2 \\ - 11 & 2 - λ \end{array} | + (- 7 - λ) | \begin{array}{c} 1 - λ & 2 \\ 3 & 2 - λ \end{array} | + 0$

Step 1.5.3

Evaluate $| \begin{array}{c} - 4 & 2 \\ - 11 & 2 - λ \end{array} |$ .

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Step 1.5.3.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = - 2 (- 4 (2 - λ) - (- 11 \cdot 2)) + (- 7 - λ) | \begin{array}{c} 1 - λ & 2 \\ 3 & 2 - λ \end{array} | + 0$

Step 1.5.3.2

Simplify the determinant.

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Step 1.5.3.2.1

Simplify each term.

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Step 1.5.3.2.1.1

Apply the distributive property.

$p (λ) = - 2 (- 4 \cdot 2 - 4 (- λ) - (- 11 \cdot 2)) + (- 7 - λ) | \begin{array}{c} 1 - λ & 2 \\ 3 & 2 - λ \end{array} | + 0$

Step 1.5.3.2.1.2

Multiply $- 4$ by $2$ .

$p (λ) = - 2 (- 8 - 4 (- λ) - (- 11 \cdot 2)) + (- 7 - λ) | \begin{array}{c} 1 - λ & 2 \\ 3 & 2 - λ \end{array} | + 0$

Step 1.5.3.2.1.3

Multiply $- 1$ by $- 4$ .

$p (λ) = - 2 (- 8 + 4 λ - (- 11 \cdot 2)) + (- 7 - λ) | \begin{array}{c} 1 - λ & 2 \\ 3 & 2 - λ \end{array} | + 0$

Step 1.5.3.2.1.4

Multiply $- (- 11 \cdot 2)$ .

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Step 1.5.3.2.1.4.1

Multiply $- 11$ by $2$ .

$p (λ) = - 2 (- 8 + 4 λ - - 22) + (- 7 - λ) | \begin{array}{c} 1 - λ & 2 \\ 3 & 2 - λ \end{array} | + 0$

Step 1.5.3.2.1.4.2

Multiply $- 1$ by $- 22$ .

$p (λ) = - 2 (- 8 + 4 λ + 22) + (- 7 - λ) | \begin{array}{c} 1 - λ & 2 \\ 3 & 2 - λ \end{array} | + 0$

Step 1.5.3.2.2

Add $- 8$ and $22$ .

$p (λ) = - 2 (4 λ + 14) + (- 7 - λ) | \begin{array}{c} 1 - λ & 2 \\ 3 & 2 - λ \end{array} | + 0$

Step 1.5.4

Evaluate $| \begin{array}{c} 1 - λ & 2 \\ 3 & 2 - λ \end{array} |$ .

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Step 1.5.4.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = - 2 (4 λ + 14) + (- 7 - λ) ((1 - λ) (2 - λ) - 3 \cdot 2) + 0$

Step 1.5.4.2

Simplify the determinant.

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Step 1.5.4.2.1

Simplify each term.

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Step 1.5.4.2.1.1

Expand $(1 - λ) (2 - λ)$ using the FOIL Method.

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Step 1.5.4.2.1.1.1

Apply the distributive property.

$p (λ) = - 2 (4 λ + 14) + (- 7 - λ) (1 (2 - λ) - λ (2 - λ) - 3 \cdot 2) + 0$

Step 1.5.4.2.1.1.2

Apply the distributive property.

$p (λ) = - 2 (4 λ + 14) + (- 7 - λ) (1 \cdot 2 + 1 (- λ) - λ (2 - λ) - 3 \cdot 2) + 0$

Step 1.5.4.2.1.1.3

Apply the distributive property.

$p (λ) = - 2 (4 λ + 14) + (- 7 - λ) (1 \cdot 2 + 1 (- λ) - λ \cdot 2 - λ (- λ) - 3 \cdot 2) + 0$

Step 1.5.4.2.1.2

Simplify and combine like terms.

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Step 1.5.4.2.1.2.1

Simplify each term.

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Step 1.5.4.2.1.2.1.1

Multiply $2$ by $1$ .

$p (λ) = - 2 (4 λ + 14) + (- 7 - λ) (2 + 1 (- λ) - λ \cdot 2 - λ (- λ) - 3 \cdot 2) + 0$

Step 1.5.4.2.1.2.1.2

Multiply $- λ$ by $1$ .

$p (λ) = - 2 (4 λ + 14) + (- 7 - λ) (2 - λ - λ \cdot 2 - λ (- λ) - 3 \cdot 2) + 0$

Step 1.5.4.2.1.2.1.3

Multiply $2$ by $- 1$ .

$p (λ) = - 2 (4 λ + 14) + (- 7 - λ) (2 - λ - 2 λ - λ (- λ) - 3 \cdot 2) + 0$

Step 1.5.4.2.1.2.1.4

Rewrite using the commutative property of multiplication.

$p (λ) = - 2 (4 λ + 14) + (- 7 - λ) (2 - λ - 2 λ - 1 \cdot - 1 λ \cdot λ - 3 \cdot 2) + 0$

Step 1.5.4.2.1.2.1.5

Multiply $λ$ by $λ$ by adding the exponents.

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Step 1.5.4.2.1.2.1.5.1

Move $λ$ .

$p (λ) = - 2 (4 λ + 14) + (- 7 - λ) (2 - λ - 2 λ - 1 \cdot - 1 (λ \cdot λ) - 3 \cdot 2) + 0$

Step 1.5.4.2.1.2.1.5.2

Multiply $λ$ by $λ$ .

$p (λ) = - 2 (4 λ + 14) + (- 7 - λ) (2 - λ - 2 λ - 1 \cdot - 1 λ^{2} - 3 \cdot 2) + 0$

Step 1.5.4.2.1.2.1.6

Multiply $- 1$ by $- 1$ .

$p (λ) = - 2 (4 λ + 14) + (- 7 - λ) (2 - λ - 2 λ + 1 λ^{2} - 3 \cdot 2) + 0$

Step 1.5.4.2.1.2.1.7

Multiply $λ^{2}$ by $1$ .

$p (λ) = - 2 (4 λ + 14) + (- 7 - λ) (2 - λ - 2 λ + λ^{2} - 3 \cdot 2) + 0$

Step 1.5.4.2.1.2.2

Subtract $2 λ$ from $- λ$ .

$p (λ) = - 2 (4 λ + 14) + (- 7 - λ) (2 - 3 λ + λ^{2} - 3 \cdot 2) + 0$

Step 1.5.4.2.1.3

Multiply $- 3$ by $2$ .

$p (λ) = - 2 (4 λ + 14) + (- 7 - λ) (2 - 3 λ + λ^{2} - 6) + 0$

Step 1.5.4.2.2

Subtract $6$ from $2$ .

$p (λ) = - 2 (4 λ + 14) + (- 7 - λ) (- 3 λ + λ^{2} - 4) + 0$

Step 1.5.4.2.3

Reorder $- 3 λ$ and $λ^{2}$ .

$p (λ) = - 2 (4 λ + 14) + (- 7 - λ) (λ^{2} - 3 λ - 4) + 0$

Step 1.5.5

Simplify the determinant.

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Step 1.5.5.1

Add $- 2 (4 λ + 14) + (- 7 - λ) (λ^{2} - 3 λ - 4)$ and $0$ .

$p (λ) = - 2 (4 λ + 14) + (- 7 - λ) (λ^{2} - 3 λ - 4)$

Step 1.5.5.2

Simplify each term.

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Step 1.5.5.2.1

Apply the distributive property.

$p (λ) = - 2 (4 λ) - 2 \cdot 14 + (- 7 - λ) (λ^{2} - 3 λ - 4)$

Step 1.5.5.2.2

Multiply $4$ by $- 2$ .

$p (λ) = - 8 λ - 2 \cdot 14 + (- 7 - λ) (λ^{2} - 3 λ - 4)$

Step 1.5.5.2.3

Multiply $- 2$ by $14$ .

$p (λ) = - 8 λ - 28 + (- 7 - λ) (λ^{2} - 3 λ - 4)$

Step 1.5.5.2.4

Expand $(- 7 - λ) (λ^{2} - 3 λ - 4)$ by multiplying each term in the first expression by each term in the second expression.

$p (λ) = - 8 λ - 28 - 7 λ^{2} - 7 (- 3 λ) - 7 \cdot - 4 - λ \cdot λ^{2} - λ (- 3 λ) - λ \cdot - 4$

Step 1.5.5.2.5

Simplify each term.

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Step 1.5.5.2.5.1

Multiply $- 3$ by $- 7$ .

$p (λ) = - 8 λ - 28 - 7 λ^{2} + 21 λ - 7 \cdot - 4 - λ \cdot λ^{2} - λ (- 3 λ) - λ \cdot - 4$

Step 1.5.5.2.5.2

Multiply $- 7$ by $- 4$ .

$p (λ) = - 8 λ - 28 - 7 λ^{2} + 21 λ + 28 - λ \cdot λ^{2} - λ (- 3 λ) - λ \cdot - 4$

Step 1.5.5.2.5.3

Multiply $λ$ by $λ^{2}$ by adding the exponents.

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Step 1.5.5.2.5.3.1

Move $λ^{2}$ .

$p (λ) = - 8 λ - 28 - 7 λ^{2} + 21 λ + 28 - (λ^{2} λ) - λ (- 3 λ) - λ \cdot - 4$

Step 1.5.5.2.5.3.2

Multiply $λ^{2}$ by $λ$ .

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Step 1.5.5.2.5.3.2.1

Raise $λ$ to the power of $1$ .

$p (λ) = - 8 λ - 28 - 7 λ^{2} + 21 λ + 28 - (λ^{2} λ^{1}) - λ (- 3 λ) - λ \cdot - 4$

Step 1.5.5.2.5.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$p (λ) = - 8 λ - 28 - 7 λ^{2} + 21 λ + 28 - λ^{2 + 1} - λ (- 3 λ) - λ \cdot - 4$

Step 1.5.5.2.5.3.3

Add $2$ and $1$ .

$p (λ) = - 8 λ - 28 - 7 λ^{2} + 21 λ + 28 - λ^{3} - λ (- 3 λ) - λ \cdot - 4$

Step 1.5.5.2.5.4

Rewrite using the commutative property of multiplication.

$p (λ) = - 8 λ - 28 - 7 λ^{2} + 21 λ + 28 - λ^{3} - 1 \cdot - 3 λ \cdot λ - λ \cdot - 4$

Step 1.5.5.2.5.5

Multiply $λ$ by $λ$ by adding the exponents.

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Step 1.5.5.2.5.5.1

Move $λ$ .

$p (λ) = - 8 λ - 28 - 7 λ^{2} + 21 λ + 28 - λ^{3} - 1 \cdot - 3 (λ \cdot λ) - λ \cdot - 4$

Step 1.5.5.2.5.5.2

Multiply $λ$ by $λ$ .

$p (λ) = - 8 λ - 28 - 7 λ^{2} + 21 λ + 28 - λ^{3} - 1 \cdot - 3 λ^{2} - λ \cdot - 4$

Step 1.5.5.2.5.6

Multiply $- 1$ by $- 3$ .

$p (λ) = - 8 λ - 28 - 7 λ^{2} + 21 λ + 28 - λ^{3} + 3 λ^{2} - λ \cdot - 4$

Step 1.5.5.2.5.7

Multiply $- 4$ by $- 1$ .

$p (λ) = - 8 λ - 28 - 7 λ^{2} + 21 λ + 28 - λ^{3} + 3 λ^{2} + 4 λ$

Step 1.5.5.2.6

Add $- 7 λ^{2}$ and $3 λ^{2}$ .

$p (λ) = - 8 λ - 28 - 4 λ^{2} + 21 λ + 28 - λ^{3} + 4 λ$

Step 1.5.5.2.7

Add $21 λ$ and $4 λ$ .

$p (λ) = - 8 λ - 28 - 4 λ^{2} + 25 λ + 28 - λ^{3}$

Step 1.5.5.3

Combine the opposite terms in $- 8 λ - 28 - 4 λ^{2} + 25 λ + 28 - λ^{3}$ .

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Step 1.5.5.3.1

Add $- 28$ and $28$ .

$p (λ) = - 8 λ - 4 λ^{2} + 25 λ + 0 - λ^{3}$

Step 1.5.5.3.2

Add $- 8 λ - 4 λ^{2} + 25 λ$ and $0$ .

$p (λ) = - 8 λ - 4 λ^{2} + 25 λ - λ^{3}$

Step 1.5.5.4

Add $- 8 λ$ and $25 λ$ .

$p (λ) = - 4 λ^{2} + 17 λ - λ^{3}$

Step 1.5.5.5

Move $17 λ$ .

$p (λ) = - 4 λ^{2} - λ^{3} + 17 λ$

Step 1.5.5.6

Reorder $- 4 λ^{2}$ and $- λ^{3}$ .

$p (λ) = - λ^{3} - 4 λ^{2} + 17 λ$

Step 1.6

Set the characteristic polynomial equal to $0$ to find the eigenvalues $λ$ .

$- λ^{3} - 4 λ^{2} + 17 λ = 0$

Step 1.7

Solve for $λ$ .

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Step 1.7.1

Factor $- λ$ out of $- λ^{3} - 4 λ^{2} + 17 λ$ .

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Step 1.7.1.1

Factor $- λ$ out of $- λ^{3}$ .

$- λ \cdot λ^{2} - 4 λ^{2} + 17 λ = 0$

Step 1.7.1.2

Factor $- λ$ out of $- 4 λ^{2}$ .

$- λ \cdot λ^{2} - λ (4 λ) + 17 λ = 0$

Step 1.7.1.3

Factor $- λ$ out of $17 λ$ .

$- λ \cdot λ^{2} - λ (4 λ) - λ \cdot - 17 = 0$

Step 1.7.1.4

Factor $- λ$ out of $- λ (λ^{2}) - λ (4 λ)$ .

$- λ (λ^{2} + 4 λ) - λ \cdot - 17 = 0$

Step 1.7.1.5

Factor $- λ$ out of $- λ (λ^{2} + 4 λ) - λ (- 17)$ .

$- λ (λ^{2} + 4 λ - 17) = 0$

Step 1.7.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$λ = 0$

$λ^{2} + 4 λ - 17 = 0$

Step 1.7.3

Set $λ$ equal to $0$ .

$λ = 0$

Step 1.7.4

Set $λ^{2} + 4 λ - 17$ equal to $0$ and solve for $λ$ .

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Step 1.7.4.1

Set $λ^{2} + 4 λ - 17$ equal to $0$ .

$λ^{2} + 4 λ - 17 = 0$

Step 1.7.4.2

Solve $λ^{2} + 4 λ - 17 = 0$ for $λ$ .

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Step 1.7.4.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 1.7.4.2.2

Substitute the values $a = 1$ , $b = 4$ , and $c = - 17$ into the quadratic formula and solve for $λ$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (1 \cdot - 17)}}{2 \cdot 1}$

Step 1.7.4.2.3

Simplify.

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Step 1.7.4.2.3.1

Simplify the numerator.

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Step 1.7.4.2.3.1.1

Raise $4$ to the power of $2$ .

$λ = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 17}}{2 \cdot 1}$

Step 1.7.4.2.3.1.2

Multiply $- 4 \cdot 1 \cdot - 17$ .

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Step 1.7.4.2.3.1.2.1

Multiply $- 4$ by $1$ .

$λ = \frac{- 4 \pm \sqrt{16 - 4 \cdot - 17}}{2 \cdot 1}$

Step 1.7.4.2.3.1.2.2

Multiply $- 4$ by $- 17$ .

$λ = \frac{- 4 \pm \sqrt{16 + 68}}{2 \cdot 1}$

Step 1.7.4.2.3.1.3

Add $16$ and $68$ .

$λ = \frac{- 4 \pm \sqrt{84}}{2 \cdot 1}$

Step 1.7.4.2.3.1.4

Rewrite $84$ as $2^{2} \cdot 21$ .

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Step 1.7.4.2.3.1.4.1

Factor $4$ out of $84$ .

$λ = \frac{- 4 \pm \sqrt{4 (21)}}{2 \cdot 1}$

Step 1.7.4.2.3.1.4.2

Rewrite $4$ as $2^{2}$ .

$λ = \frac{- 4 \pm \sqrt{2^{2} \cdot 21}}{2 \cdot 1}$

Step 1.7.4.2.3.1.5

Pull terms out from under the radical.

$λ = \frac{- 4 \pm 2 \sqrt{21}}{2 \cdot 1}$

Step 1.7.4.2.3.2

Multiply $2$ by $1$ .

$λ = \frac{- 4 \pm 2 \sqrt{21}}{2}$

Step 1.7.4.2.3.3

Simplify $\frac{- 4 \pm 2 \sqrt{21}}{2}$ .

$λ = - 2 \pm \sqrt{21}$

Step 1.7.4.2.4

The final answer is the combination of both solutions.

$λ = - 2 + \sqrt{21}, - 2 - \sqrt{21}$

Step 1.7.5

The final solution is all the values that make $- λ (λ^{2} + 4 λ - 17) = 0$ true.

$λ = 0, - 2 + \sqrt{21}, - 2 - \sqrt{21}$

Step 2

The eigenvector is equal to the null space of the matrix minus the eigenvalue times the identity matrix where $N$ is the null space and $I$ is the identity matrix.

$ε_{A} = N (A - λ I_{3})$

Step 3

Find the eigenvector using the eigenvalue $λ = 0$ .

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Step 3.1

Substitute the known values into the formula.

$N ([\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + 0 [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}])$

Step 3.2

Simplify.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Multiply $0$ by each element of the matrix.

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 0 \cdot 1 & 0 \cdot 0 & 0 \cdot 0 \\ 0 \cdot 0 & 0 \cdot 1 & 0 \cdot 0 \\ 0 \cdot 0 & 0 \cdot 0 & 0 \cdot 1 \end{array}]$

Step 3.2.1.2

Simplify each element in the matrix.

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Step 3.2.1.2.1

Multiply $0$ by $1$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 0 & 0 \cdot 0 & 0 \cdot 0 \\ 0 \cdot 0 & 0 \cdot 1 & 0 \cdot 0 \\ 0 \cdot 0 & 0 \cdot 0 & 0 \cdot 1 \end{array}]$

Step 3.2.1.2.2

Multiply $0$ by $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 0 & 0 & 0 \cdot 0 \\ 0 \cdot 0 & 0 \cdot 1 & 0 \cdot 0 \\ 0 \cdot 0 & 0 \cdot 0 & 0 \cdot 1 \end{array}]$

Step 3.2.1.2.3

Multiply $0$ by $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 0 & 0 & 0 \\ 0 \cdot 0 & 0 \cdot 1 & 0 \cdot 0 \\ 0 \cdot 0 & 0 \cdot 0 & 0 \cdot 1 \end{array}]$

Step 3.2.1.2.4

Multiply $0$ by $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 \cdot 1 & 0 \cdot 0 \\ 0 \cdot 0 & 0 \cdot 0 & 0 \cdot 1 \end{array}]$

Step 3.2.1.2.5

Multiply $0$ by $1$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \cdot 0 \\ 0 \cdot 0 & 0 \cdot 0 & 0 \cdot 1 \end{array}]$

Step 3.2.1.2.6

Multiply $0$ by $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 \cdot 0 & 0 \cdot 0 & 0 \cdot 1 \end{array}]$

Step 3.2.1.2.7

Multiply $0$ by $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 \cdot 0 & 0 \cdot 1 \end{array}]$

Step 3.2.1.2.8

Multiply $0$ by $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \cdot 1 \end{array}]$

Step 3.2.1.2.9

Multiply $0$ by $1$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}]$

Step 3.2.2

Adding any matrix to a null matrix is the matrix itself.

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Step 3.2.2.1

Add the corresponding elements.

$[\begin{array}{c} 1 + 0 & - 4 + 0 & 2 + 0 \\ 2 + 0 & - 7 + 0 & 0 + 0 \\ 3 + 0 & - 11 + 0 & 2 + 0 \end{array}]$

Step 3.2.2.2

Simplify each element.

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Step 3.2.2.2.1

Add $1$ and $0$ .

$[\begin{array}{c} 1 & - 4 + 0 & 2 + 0 \\ 2 + 0 & - 7 + 0 & 0 + 0 \\ 3 + 0 & - 11 + 0 & 2 + 0 \end{array}]$

Step 3.2.2.2.2

Add $- 4$ and $0$ .

$[\begin{array}{c} 1 & - 4 & 2 + 0 \\ 2 + 0 & - 7 + 0 & 0 + 0 \\ 3 + 0 & - 11 + 0 & 2 + 0 \end{array}]$

Step 3.2.2.2.3

Add $2$ and $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 + 0 & - 7 + 0 & 0 + 0 \\ 3 + 0 & - 11 + 0 & 2 + 0 \end{array}]$

Step 3.2.2.2.4

Add $2$ and $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 + 0 & 0 + 0 \\ 3 + 0 & - 11 + 0 & 2 + 0 \end{array}]$

Step 3.2.2.2.5

Add $- 7$ and $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 + 0 \\ 3 + 0 & - 11 + 0 & 2 + 0 \end{array}]$

Step 3.2.2.2.6

Add $0$ and $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 + 0 & - 11 + 0 & 2 + 0 \end{array}]$

Step 3.2.2.2.7

Add $3$ and $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 + 0 & 2 + 0 \end{array}]$

Step 3.2.2.2.8

Add $- 11$ and $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 + 0 \end{array}]$

Step 3.2.2.2.9

Add $2$ and $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}]$

Step 3.3

Find the null space when $λ = 0$ .

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Step 3.3.1

Write as an augmented matrix for $A x = 0$ .

$[\begin{array}{c} 1 & - 4 & 2 & 0 \\ 2 & - 7 & 0 & 0 \\ 3 & - 11 & 2 & 0 \end{array}]$

Step 3.3.2

Find the reduced row echelon form.

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Step 3.3.2.1

Perform the row operation $R_{2} = R_{2} - 2 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 3.3.2.1.1

Perform the row operation $R_{2} = R_{2} - 2 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & - 4 & 2 & 0 \\ 2 - 2 \cdot 1 & - 7 - 2 \cdot - 4 & 0 - 2 \cdot 2 & 0 - 2 \cdot 0 \\ 3 & - 11 & 2 & 0 \end{array}]$

Step 3.3.2.1.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - 4 & 2 & 0 \\ 0 & 1 & - 4 & 0 \\ 3 & - 11 & 2 & 0 \end{array}]$

Step 3.3.2.2

Perform the row operation $R_{3} = R_{3} - 3 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Step 3.3.2.2.1

Perform the row operation $R_{3} = R_{3} - 3 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & - 4 & 2 & 0 \\ 0 & 1 & - 4 & 0 \\ 3 - 3 \cdot 1 & - 11 - 3 \cdot - 4 & 2 - 3 \cdot 2 & 0 - 3 \cdot 0 \end{array}]$

Step 3.3.2.2.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - 4 & 2 & 0 \\ 0 & 1 & - 4 & 0 \\ 0 & 1 & - 4 & 0 \end{array}]$

Step 3.3.2.3

Perform the row operation $R_{3} = R_{3} - R_{2}$ to make the entry at $3, 2$ a $0$ .

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Step 3.3.2.3.1

Perform the row operation $R_{3} = R_{3} - R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & - 4 & 2 & 0 \\ 0 & 1 & - 4 & 0 \\ 0 - 0 & 1 - 1 & - 4 + 4 & 0 - 0 \end{array}]$

Step 3.3.2.3.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - 4 & 2 & 0 \\ 0 & 1 & - 4 & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Step 3.3.2.4

Perform the row operation $R_{1} = R_{1} + 4 R_{2}$ to make the entry at $1, 2$ a $0$ .

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Step 3.3.2.4.1

Perform the row operation $R_{1} = R_{1} + 4 R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 + 4 \cdot 0 & - 4 + 4 \cdot 1 & 2 + 4 \cdot - 4 & 0 + 4 \cdot 0 \\ 0 & 1 & - 4 & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Step 3.3.2.4.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & - 14 & 0 \\ 0 & 1 & - 4 & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Step 3.3.3

Use the result matrix to declare the final solution to the system of equations.

$x - 14 z = 0$

$y - 4 z = 0$

$0 = 0$

Step 3.3.4

Write a solution vector by solving in terms of the free variables in each row.

$[\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 14 z \\ 4 z \\ z \end{array}]$

Step 3.3.5

Write the solution as a linear combination of vectors.

$[\begin{array}{c} x \\ y \\ z \end{array}] = z [\begin{array}{c} 14 \\ 4 \\ 1 \end{array}]$

Step 3.3.6

Write as a solution set.

${z [\begin{array}{c} 14 \\ 4 \\ 1 \end{array}] | z \in R}$

Step 3.3.7

The solution is the set of vectors created from the free variables of the system.

${[\begin{array}{c} 14 \\ 4 \\ 1 \end{array}]}$

Step 4

Find the eigenvector using the eigenvalue $λ = - 2 + \sqrt{21}$ .

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Step 4.1

Substitute the known values into the formula.

$N ([\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] - (- 2 + \sqrt{21}) [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}])$

Step 4.2

Simplify.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Apply the distributive property.

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + (- - 2 - \sqrt{21}) [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Step 4.2.1.2

Multiply $- 1$ by $- 2$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + (2 - \sqrt{21}) [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Step 4.2.1.3

Multiply $2 - \sqrt{21}$ by each element of the matrix.

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} (2 - \sqrt{21}) \cdot 1 & (2 - \sqrt{21}) \cdot 0 & (2 - \sqrt{21}) \cdot 0 \\ (2 - \sqrt{21}) \cdot 0 & (2 - \sqrt{21}) \cdot 1 & (2 - \sqrt{21}) \cdot 0 \\ (2 - \sqrt{21}) \cdot 0 & (2 - \sqrt{21}) \cdot 0 & (2 - \sqrt{21}) \cdot 1 \end{array}]$

Step 4.2.1.4

Simplify each element in the matrix.

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Step 4.2.1.4.1

Multiply $2 - \sqrt{21}$ by $1$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 2 - \sqrt{21} & (2 - \sqrt{21}) \cdot 0 & (2 - \sqrt{21}) \cdot 0 \\ (2 - \sqrt{21}) \cdot 0 & (2 - \sqrt{21}) \cdot 1 & (2 - \sqrt{21}) \cdot 0 \\ (2 - \sqrt{21}) \cdot 0 & (2 - \sqrt{21}) \cdot 0 & (2 - \sqrt{21}) \cdot 1 \end{array}]$

Step 4.2.1.4.2

Multiply $2 - \sqrt{21}$ by $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 2 - \sqrt{21} & 0 & (2 - \sqrt{21}) \cdot 0 \\ (2 - \sqrt{21}) \cdot 0 & (2 - \sqrt{21}) \cdot 1 & (2 - \sqrt{21}) \cdot 0 \\ (2 - \sqrt{21}) \cdot 0 & (2 - \sqrt{21}) \cdot 0 & (2 - \sqrt{21}) \cdot 1 \end{array}]$

Step 4.2.1.4.3

Multiply $2 - \sqrt{21}$ by $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 2 - \sqrt{21} & 0 & 0 \\ (2 - \sqrt{21}) \cdot 0 & (2 - \sqrt{21}) \cdot 1 & (2 - \sqrt{21}) \cdot 0 \\ (2 - \sqrt{21}) \cdot 0 & (2 - \sqrt{21}) \cdot 0 & (2 - \sqrt{21}) \cdot 1 \end{array}]$

Step 4.2.1.4.4

Multiply $2 - \sqrt{21}$ by $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 2 - \sqrt{21} & 0 & 0 \\ 0 & (2 - \sqrt{21}) \cdot 1 & (2 - \sqrt{21}) \cdot 0 \\ (2 - \sqrt{21}) \cdot 0 & (2 - \sqrt{21}) \cdot 0 & (2 - \sqrt{21}) \cdot 1 \end{array}]$

Step 4.2.1.4.5

Multiply $2 - \sqrt{21}$ by $1$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 2 - \sqrt{21} & 0 & 0 \\ 0 & 2 - \sqrt{21} & (2 - \sqrt{21}) \cdot 0 \\ (2 - \sqrt{21}) \cdot 0 & (2 - \sqrt{21}) \cdot 0 & (2 - \sqrt{21}) \cdot 1 \end{array}]$

Step 4.2.1.4.6

Multiply $2 - \sqrt{21}$ by $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 2 - \sqrt{21} & 0 & 0 \\ 0 & 2 - \sqrt{21} & 0 \\ (2 - \sqrt{21}) \cdot 0 & (2 - \sqrt{21}) \cdot 0 & (2 - \sqrt{21}) \cdot 1 \end{array}]$

Step 4.2.1.4.7

Multiply $2 - \sqrt{21}$ by $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 2 - \sqrt{21} & 0 & 0 \\ 0 & 2 - \sqrt{21} & 0 \\ 0 & (2 - \sqrt{21}) \cdot 0 & (2 - \sqrt{21}) \cdot 1 \end{array}]$

Step 4.2.1.4.8

Multiply $2 - \sqrt{21}$ by $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 2 - \sqrt{21} & 0 & 0 \\ 0 & 2 - \sqrt{21} & 0 \\ 0 & 0 & (2 - \sqrt{21}) \cdot 1 \end{array}]$

Step 4.2.1.4.9

Multiply $2 - \sqrt{21}$ by $1$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 2 - \sqrt{21} & 0 & 0 \\ 0 & 2 - \sqrt{21} & 0 \\ 0 & 0 & 2 - \sqrt{21} \end{array}]$

Step 4.2.2

Add the corresponding elements.

$[\begin{array}{c} 1 + 2 - \sqrt{21} & - 4 + 0 & 2 + 0 \\ 2 + 0 & - 7 + 2 - \sqrt{21} & 0 + 0 \\ 3 + 0 & - 11 + 0 & 2 + 2 - \sqrt{21} \end{array}]$

Step 4.2.3

Simplify each element.

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Step 4.2.3.1

Add $1$ and $2$ .

$[\begin{array}{c} 3 - \sqrt{21} & - 4 + 0 & 2 + 0 \\ 2 + 0 & - 7 + 2 - \sqrt{21} & 0 + 0 \\ 3 + 0 & - 11 + 0 & 2 + 2 - \sqrt{21} \end{array}]$

Step 4.2.3.2

Add $- 4$ and $0$ .

$[\begin{array}{c} 3 - \sqrt{21} & - 4 & 2 + 0 \\ 2 + 0 & - 7 + 2 - \sqrt{21} & 0 + 0 \\ 3 + 0 & - 11 + 0 & 2 + 2 - \sqrt{21} \end{array}]$

Step 4.2.3.3

Add $2$ and $0$ .

$[\begin{array}{c} 3 - \sqrt{21} & - 4 & 2 \\ 2 + 0 & - 7 + 2 - \sqrt{21} & 0 + 0 \\ 3 + 0 & - 11 + 0 & 2 + 2 - \sqrt{21} \end{array}]$

Step 4.2.3.4

Add $2$ and $0$ .

$[\begin{array}{c} 3 - \sqrt{21} & - 4 & 2 \\ 2 & - 7 + 2 - \sqrt{21} & 0 + 0 \\ 3 + 0 & - 11 + 0 & 2 + 2 - \sqrt{21} \end{array}]$

Step 4.2.3.5

Add $- 7$ and $2$ .

$[\begin{array}{c} 3 - \sqrt{21} & - 4 & 2 \\ 2 & - 5 - \sqrt{21} & 0 + 0 \\ 3 + 0 & - 11 + 0 & 2 + 2 - \sqrt{21} \end{array}]$

Step 4.2.3.6

Add $0$ and $0$ .

$[\begin{array}{c} 3 - \sqrt{21} & - 4 & 2 \\ 2 & - 5 - \sqrt{21} & 0 \\ 3 + 0 & - 11 + 0 & 2 + 2 - \sqrt{21} \end{array}]$

Step 4.2.3.7

Add $3$ and $0$ .

$[\begin{array}{c} 3 - \sqrt{21} & - 4 & 2 \\ 2 & - 5 - \sqrt{21} & 0 \\ 3 & - 11 + 0 & 2 + 2 - \sqrt{21} \end{array}]$

Step 4.2.3.8

Add $- 11$ and $0$ .

$[\begin{array}{c} 3 - \sqrt{21} & - 4 & 2 \\ 2 & - 5 - \sqrt{21} & 0 \\ 3 & - 11 & 2 + 2 - \sqrt{21} \end{array}]$

Step 4.2.3.9

Add $2$ and $2$ .

$[\begin{array}{c} 3 - \sqrt{21} & - 4 & 2 \\ 2 & - 5 - \sqrt{21} & 0 \\ 3 & - 11 & 4 - \sqrt{21} \end{array}]$

Step 4.3

Find the null space when $λ = - 2 + \sqrt{21}$ .

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Step 4.3.1

Write as an augmented matrix for $A x = 0$ .

$[\begin{array}{c} 3 - \sqrt{21} & - 4 & 2 & 0 \\ 2 & - 5 - \sqrt{21} & 0 & 0 \\ 3 & - 11 & 4 - \sqrt{21} & 0 \end{array}]$

Step 4.3.2

Find the reduced row echelon form.

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Step 4.3.2.1

Multiply each element of $R_{1}$ by $\frac{1}{3 - \sqrt{21}}$ to make the entry at $1, 1$ a $1$ .

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Step 4.3.2.1.1

Multiply each element of $R_{1}$ by $\frac{1}{3 - \sqrt{21}}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{3 - \sqrt{21}}{3 - \sqrt{21}} & \frac{- 4}{3 - \sqrt{21}} & \frac{2}{3 - \sqrt{21}} & \frac{0}{3 - \sqrt{21}} \\ 2 & - 5 - \sqrt{21} & 0 & 0 \\ 3 & - 11 & 4 - \sqrt{21} & 0 \end{array}]$

Step 4.3.2.1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & \frac{3 + \sqrt{21}}{3} & - \frac{3 + \sqrt{21}}{6} & 0 \\ 2 & - 5 - \sqrt{21} & 0 & 0 \\ 3 & - 11 & 4 - \sqrt{21} & 0 \end{array}]$

Step 4.3.2.2

Perform the row operation $R_{2} = R_{2} - 2 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 4.3.2.2.1

Perform the row operation $R_{2} = R_{2} - 2 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{3 + \sqrt{21}}{3} & - \frac{3 + \sqrt{21}}{6} & 0 \\ 2 - 2 \cdot 1 & - 5 - \sqrt{21} - 2 \frac{3 + \sqrt{21}}{3} & 0 - 2 (- \frac{3 + \sqrt{21}}{6}) & 0 - 2 \cdot 0 \\ 3 & - 11 & 4 - \sqrt{21} & 0 \end{array}]$

Step 4.3.2.2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & \frac{3 + \sqrt{21}}{3} & - \frac{3 + \sqrt{21}}{6} & 0 \\ 0 & - \frac{21 + 5 \sqrt{21}}{3} & \frac{3 + \sqrt{21}}{3} & 0 \\ 3 & - 11 & 4 - \sqrt{21} & 0 \end{array}]$

Step 4.3.2.3

Perform the row operation $R_{3} = R_{3} - 3 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Step 4.3.2.3.1

Perform the row operation $R_{3} = R_{3} - 3 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{3 + \sqrt{21}}{3} & - \frac{3 + \sqrt{21}}{6} & 0 \\ 0 & - \frac{21 + 5 \sqrt{21}}{3} & \frac{3 + \sqrt{21}}{3} & 0 \\ 3 - 3 \cdot 1 & - 11 - 3 \frac{3 + \sqrt{21}}{3} & 4 - \sqrt{21} - 3 (- \frac{3 + \sqrt{21}}{6}) & 0 - 3 \cdot 0 \end{array}]$

Step 4.3.2.3.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & \frac{3 + \sqrt{21}}{3} & - \frac{3 + \sqrt{21}}{6} & 0 \\ 0 & - \frac{21 + 5 \sqrt{21}}{3} & \frac{3 + \sqrt{21}}{3} & 0 \\ 0 & - 14 - \sqrt{21} & \frac{11 - \sqrt{21}}{2} & 0 \end{array}]$

Step 4.3.2.4

Multiply each element of $R_{2}$ by $- \frac{3}{21 + 5 \sqrt{21}}$ to make the entry at $2, 2$ a $1$ .

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Step 4.3.2.4.1

Multiply each element of $R_{2}$ by $- \frac{3}{21 + 5 \sqrt{21}}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{3 + \sqrt{21}}{3} & - \frac{3 + \sqrt{21}}{6} & 0 \\ - \frac{3}{21 + 5 \sqrt{21}} \cdot 0 & - \frac{3}{21 + 5 \sqrt{21}} (- \frac{21 + 5 \sqrt{21}}{3}) & - \frac{3}{21 + 5 \sqrt{21}} \cdot \frac{3 + \sqrt{21}}{3} & - \frac{3}{21 + 5 \sqrt{21}} \cdot 0 \\ 0 & - 14 - \sqrt{21} & \frac{11 - \sqrt{21}}{2} & 0 \end{array}]$

Step 4.3.2.4.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & \frac{3 + \sqrt{21}}{3} & - \frac{3 + \sqrt{21}}{6} & 0 \\ 0 & 1 & - \frac{7 - \sqrt{21}}{14} & 0 \\ 0 & - 14 - \sqrt{21} & \frac{11 - \sqrt{21}}{2} & 0 \end{array}]$

Step 4.3.2.5

Perform the row operation $R_{3} = R_{3} - (- 14 - \sqrt{21}) R_{2}$ to make the entry at $3, 2$ a $0$ .

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Step 4.3.2.5.1

Perform the row operation $R_{3} = R_{3} - (- 14 - \sqrt{21}) R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{3 + \sqrt{21}}{3} & - \frac{3 + \sqrt{21}}{6} & 0 \\ 0 & 1 & - \frac{7 - \sqrt{21}}{14} & 0 \\ 0 - (- 14 - \sqrt{21}) \cdot 0 & - 14 - \sqrt{21} - (- 14 - \sqrt{21}) \cdot 1 & \frac{11 - \sqrt{21}}{2} - (- 14 - \sqrt{21}) (- \frac{7 - \sqrt{21}}{14}) & 0 - (- 14 - \sqrt{21}) \cdot 0 \end{array}]$

Step 4.3.2.5.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & \frac{3 + \sqrt{21}}{3} & - \frac{3 + \sqrt{21}}{6} & 0 \\ 0 & 1 & - \frac{7 - \sqrt{21}}{14} & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Step 4.3.2.6

Perform the row operation $R_{1} = R_{1} - \frac{3 + \sqrt{21}}{3} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Step 4.3.2.6.1

Perform the row operation $R_{1} = R_{1} - \frac{3 + \sqrt{21}}{3} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - \frac{3 + \sqrt{21}}{3} \cdot 0 & \frac{3 + \sqrt{21}}{3} - \frac{3 + \sqrt{21}}{3} \cdot 1 & - \frac{3 + \sqrt{21}}{6} - \frac{3 + \sqrt{21}}{3} (- \frac{7 - \sqrt{21}}{14}) & 0 - \frac{3 + \sqrt{21}}{3} \cdot 0 \\ 0 & 1 & - \frac{7 - \sqrt{21}}{14} & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Step 4.3.2.6.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & - \frac{7 + \sqrt{21}}{14} & 0 \\ 0 & 1 & - \frac{7 - \sqrt{21}}{14} & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Step 4.3.3

Use the result matrix to declare the final solution to the system of equations.

$x - \frac{7 + \sqrt{21}}{14} z = 0$

$y - \frac{7 - \sqrt{21}}{14} z = 0$

$0 = 0$

Step 4.3.4

Write a solution vector by solving in terms of the free variables in each row.

$[\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} \frac{z}{2} + \frac{z \sqrt{21}}{14} \\ \frac{z}{2} - \frac{z \sqrt{21}}{14} \\ z \end{array}]$

Step 4.3.5

Write the solution as a linear combination of vectors.

$[\begin{array}{c} x \\ y \\ z \end{array}] = z [\begin{array}{c} \frac{1}{2} + \frac{\sqrt{21}}{14} \\ \frac{1}{2} - \frac{\sqrt{21}}{14} \\ 1 \end{array}]$

Step 4.3.6

Write as a solution set.

${z [\begin{array}{c} \frac{1}{2} + \frac{\sqrt{21}}{14} \\ \frac{1}{2} - \frac{\sqrt{21}}{14} \\ 1 \end{array}] | z \in R}$

Step 4.3.7

The solution is the set of vectors created from the free variables of the system.

${[\begin{array}{c} \frac{1}{2} + \frac{\sqrt{21}}{14} \\ \frac{1}{2} - \frac{\sqrt{21}}{14} \\ 1 \end{array}]}$

Step 5

Find the eigenvector using the eigenvalue $λ = - 2 - \sqrt{21}$ .

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Step 5.1

Substitute the known values into the formula.

$N ([\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] - (- 2 - \sqrt{21}) [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}])$

Step 5.2

Simplify.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Apply the distributive property.

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + (- - 2 - - \sqrt{21}) [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Step 5.2.1.2

Multiply $- 1$ by $- 2$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + (2 - - \sqrt{21}) [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Step 5.2.1.3

Multiply $- - \sqrt{21}$ .

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Step 5.2.1.3.1

Multiply $- 1$ by $- 1$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + (2 + 1 \sqrt{21}) [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Step 5.2.1.3.2

Multiply $\sqrt{21}$ by $1$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + (2 + \sqrt{21}) [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Step 5.2.1.4

Multiply $2 + \sqrt{21}$ by each element of the matrix.

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} (2 + \sqrt{21}) \cdot 1 & (2 + \sqrt{21}) \cdot 0 & (2 + \sqrt{21}) \cdot 0 \\ (2 + \sqrt{21}) \cdot 0 & (2 + \sqrt{21}) \cdot 1 & (2 + \sqrt{21}) \cdot 0 \\ (2 + \sqrt{21}) \cdot 0 & (2 + \sqrt{21}) \cdot 0 & (2 + \sqrt{21}) \cdot 1 \end{array}]$

Step 5.2.1.5

Simplify each element in the matrix.

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Step 5.2.1.5.1

Multiply $2 + \sqrt{21}$ by $1$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 2 + \sqrt{21} & (2 + \sqrt{21}) \cdot 0 & (2 + \sqrt{21}) \cdot 0 \\ (2 + \sqrt{21}) \cdot 0 & (2 + \sqrt{21}) \cdot 1 & (2 + \sqrt{21}) \cdot 0 \\ (2 + \sqrt{21}) \cdot 0 & (2 + \sqrt{21}) \cdot 0 & (2 + \sqrt{21}) \cdot 1 \end{array}]$

Step 5.2.1.5.2

Multiply $2 + \sqrt{21}$ by $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 2 + \sqrt{21} & 0 & (2 + \sqrt{21}) \cdot 0 \\ (2 + \sqrt{21}) \cdot 0 & (2 + \sqrt{21}) \cdot 1 & (2 + \sqrt{21}) \cdot 0 \\ (2 + \sqrt{21}) \cdot 0 & (2 + \sqrt{21}) \cdot 0 & (2 + \sqrt{21}) \cdot 1 \end{array}]$

Step 5.2.1.5.3

Multiply $2 + \sqrt{21}$ by $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 2 + \sqrt{21} & 0 & 0 \\ (2 + \sqrt{21}) \cdot 0 & (2 + \sqrt{21}) \cdot 1 & (2 + \sqrt{21}) \cdot 0 \\ (2 + \sqrt{21}) \cdot 0 & (2 + \sqrt{21}) \cdot 0 & (2 + \sqrt{21}) \cdot 1 \end{array}]$

Step 5.2.1.5.4

Multiply $2 + \sqrt{21}$ by $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 2 + \sqrt{21} & 0 & 0 \\ 0 & (2 + \sqrt{21}) \cdot 1 & (2 + \sqrt{21}) \cdot 0 \\ (2 + \sqrt{21}) \cdot 0 & (2 + \sqrt{21}) \cdot 0 & (2 + \sqrt{21}) \cdot 1 \end{array}]$

Step 5.2.1.5.5

Multiply $2 + \sqrt{21}$ by $1$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 2 + \sqrt{21} & 0 & 0 \\ 0 & 2 + \sqrt{21} & (2 + \sqrt{21}) \cdot 0 \\ (2 + \sqrt{21}) \cdot 0 & (2 + \sqrt{21}) \cdot 0 & (2 + \sqrt{21}) \cdot 1 \end{array}]$

Step 5.2.1.5.6

Multiply $2 + \sqrt{21}$ by $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 2 + \sqrt{21} & 0 & 0 \\ 0 & 2 + \sqrt{21} & 0 \\ (2 + \sqrt{21}) \cdot 0 & (2 + \sqrt{21}) \cdot 0 & (2 + \sqrt{21}) \cdot 1 \end{array}]$

Step 5.2.1.5.7

Multiply $2 + \sqrt{21}$ by $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 2 + \sqrt{21} & 0 & 0 \\ 0 & 2 + \sqrt{21} & 0 \\ 0 & (2 + \sqrt{21}) \cdot 0 & (2 + \sqrt{21}) \cdot 1 \end{array}]$

Step 5.2.1.5.8

Multiply $2 + \sqrt{21}$ by $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 2 + \sqrt{21} & 0 & 0 \\ 0 & 2 + \sqrt{21} & 0 \\ 0 & 0 & (2 + \sqrt{21}) \cdot 1 \end{array}]$

Step 5.2.1.5.9

Multiply $2 + \sqrt{21}$ by $1$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 2 + \sqrt{21} & 0 & 0 \\ 0 & 2 + \sqrt{21} & 0 \\ 0 & 0 & 2 + \sqrt{21} \end{array}]$

Step 5.2.2

Add the corresponding elements.

$[\begin{array}{c} 1 + 2 + \sqrt{21} & - 4 + 0 & 2 + 0 \\ 2 + 0 & - 7 + 2 + \sqrt{21} & 0 + 0 \\ 3 + 0 & - 11 + 0 & 2 + 2 + \sqrt{21} \end{array}]$

Step 5.2.3

Simplify each element.

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Step 5.2.3.1

Add $1$ and $2$ .

$[\begin{array}{c} 3 + \sqrt{21} & - 4 + 0 & 2 + 0 \\ 2 + 0 & - 7 + 2 + \sqrt{21} & 0 + 0 \\ 3 + 0 & - 11 + 0 & 2 + 2 + \sqrt{21} \end{array}]$

Step 5.2.3.2

Add $- 4$ and $0$ .

$[\begin{array}{c} 3 + \sqrt{21} & - 4 & 2 + 0 \\ 2 + 0 & - 7 + 2 + \sqrt{21} & 0 + 0 \\ 3 + 0 & - 11 + 0 & 2 + 2 + \sqrt{21} \end{array}]$

Step 5.2.3.3

Add $2$ and $0$ .

$[\begin{array}{c} 3 + \sqrt{21} & - 4 & 2 \\ 2 + 0 & - 7 + 2 + \sqrt{21} & 0 + 0 \\ 3 + 0 & - 11 + 0 & 2 + 2 + \sqrt{21} \end{array}]$

Step 5.2.3.4

Add $2$ and $0$ .

$[\begin{array}{c} 3 + \sqrt{21} & - 4 & 2 \\ 2 & - 7 + 2 + \sqrt{21} & 0 + 0 \\ 3 + 0 & - 11 + 0 & 2 + 2 + \sqrt{21} \end{array}]$

Step 5.2.3.5

Add $- 7$ and $2$ .

$[\begin{array}{c} 3 + \sqrt{21} & - 4 & 2 \\ 2 & - 5 + \sqrt{21} & 0 + 0 \\ 3 + 0 & - 11 + 0 & 2 + 2 + \sqrt{21} \end{array}]$

Step 5.2.3.6

Add $0$ and $0$ .

$[\begin{array}{c} 3 + \sqrt{21} & - 4 & 2 \\ 2 & - 5 + \sqrt{21} & 0 \\ 3 + 0 & - 11 + 0 & 2 + 2 + \sqrt{21} \end{array}]$

Step 5.2.3.7

Add $3$ and $0$ .

$[\begin{array}{c} 3 + \sqrt{21} & - 4 & 2 \\ 2 & - 5 + \sqrt{21} & 0 \\ 3 & - 11 + 0 & 2 + 2 + \sqrt{21} \end{array}]$

Step 5.2.3.8

Add $- 11$ and $0$ .

$[\begin{array}{c} 3 + \sqrt{21} & - 4 & 2 \\ 2 & - 5 + \sqrt{21} & 0 \\ 3 & - 11 & 2 + 2 + \sqrt{21} \end{array}]$

Step 5.2.3.9

Add $2$ and $2$ .

$[\begin{array}{c} 3 + \sqrt{21} & - 4 & 2 \\ 2 & - 5 + \sqrt{21} & 0 \\ 3 & - 11 & 4 + \sqrt{21} \end{array}]$

Step 5.3

Find the null space when $λ = - 2 - \sqrt{21}$ .

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Step 5.3.1

Write as an augmented matrix for $A x = 0$ .

$[\begin{array}{c} 3 + \sqrt{21} & - 4 & 2 & 0 \\ 2 & - 5 + \sqrt{21} & 0 & 0 \\ 3 & - 11 & 4 + \sqrt{21} & 0 \end{array}]$

Step 5.3.2

Find the reduced row echelon form.

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Step 5.3.2.1

Multiply each element of $R_{1}$ by $\frac{1}{3 + \sqrt{21}}$ to make the entry at $1, 1$ a $1$ .

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Step 5.3.2.1.1

Multiply each element of $R_{1}$ by $\frac{1}{3 + \sqrt{21}}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{3 + \sqrt{21}}{3 + \sqrt{21}} & \frac{- 4}{3 + \sqrt{21}} & \frac{2}{3 + \sqrt{21}} & \frac{0}{3 + \sqrt{21}} \\ 2 & - 5 + \sqrt{21} & 0 & 0 \\ 3 & - 11 & 4 + \sqrt{21} & 0 \end{array}]$

Step 5.3.2.1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & \frac{3 - \sqrt{21}}{3} & - \frac{3 - \sqrt{21}}{6} & 0 \\ 2 & - 5 + \sqrt{21} & 0 & 0 \\ 3 & - 11 & 4 + \sqrt{21} & 0 \end{array}]$

Step 5.3.2.2

Perform the row operation $R_{2} = R_{2} - 2 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 5.3.2.2.1

Perform the row operation $R_{2} = R_{2} - 2 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{3 - \sqrt{21}}{3} & - \frac{3 - \sqrt{21}}{6} & 0 \\ 2 - 2 \cdot 1 & - 5 + \sqrt{21} - 2 \frac{3 - \sqrt{21}}{3} & 0 - 2 (- \frac{3 - \sqrt{21}}{6}) & 0 - 2 \cdot 0 \\ 3 & - 11 & 4 + \sqrt{21} & 0 \end{array}]$

Step 5.3.2.2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & \frac{3 - \sqrt{21}}{3} & - \frac{3 - \sqrt{21}}{6} & 0 \\ 0 & - \frac{21 - 5 \sqrt{21}}{3} & \frac{3 - \sqrt{21}}{3} & 0 \\ 3 & - 11 & 4 + \sqrt{21} & 0 \end{array}]$

Step 5.3.2.3

Perform the row operation $R_{3} = R_{3} - 3 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Step 5.3.2.3.1

Perform the row operation $R_{3} = R_{3} - 3 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{3 - \sqrt{21}}{3} & - \frac{3 - \sqrt{21}}{6} & 0 \\ 0 & - \frac{21 - 5 \sqrt{21}}{3} & \frac{3 - \sqrt{21}}{3} & 0 \\ 3 - 3 \cdot 1 & - 11 - 3 \frac{3 - \sqrt{21}}{3} & 4 + \sqrt{21} - 3 (- \frac{3 - \sqrt{21}}{6}) & 0 - 3 \cdot 0 \end{array}]$

Step 5.3.2.3.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & \frac{3 - \sqrt{21}}{3} & - \frac{3 - \sqrt{21}}{6} & 0 \\ 0 & - \frac{21 - 5 \sqrt{21}}{3} & \frac{3 - \sqrt{21}}{3} & 0 \\ 0 & - 14 + \sqrt{21} & \frac{11 + \sqrt{21}}{2} & 0 \end{array}]$

Step 5.3.2.4

Multiply each element of $R_{2}$ by $- \frac{3}{21 - 5 \sqrt{21}}$ to make the entry at $2, 2$ a $1$ .

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Step 5.3.2.4.1

Multiply each element of $R_{2}$ by $- \frac{3}{21 - 5 \sqrt{21}}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{3 - \sqrt{21}}{3} & - \frac{3 - \sqrt{21}}{6} & 0 \\ - \frac{3}{21 - 5 \sqrt{21}} \cdot 0 & - \frac{3}{21 - 5 \sqrt{21}} (- \frac{21 - 5 \sqrt{21}}{3}) & - \frac{3}{21 - 5 \sqrt{21}} \cdot \frac{3 - \sqrt{21}}{3} & - \frac{3}{21 - 5 \sqrt{21}} \cdot 0 \\ 0 & - 14 + \sqrt{21} & \frac{11 + \sqrt{21}}{2} & 0 \end{array}]$

Step 5.3.2.4.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & \frac{3 - \sqrt{21}}{3} & - \frac{3 - \sqrt{21}}{6} & 0 \\ 0 & 1 & - \frac{7 + \sqrt{21}}{14} & 0 \\ 0 & - 14 + \sqrt{21} & \frac{11 + \sqrt{21}}{2} & 0 \end{array}]$

Step 5.3.2.5

Perform the row operation $R_{3} = R_{3} - (- 14 + \sqrt{21}) R_{2}$ to make the entry at $3, 2$ a $0$ .

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Step 5.3.2.5.1

Perform the row operation $R_{3} = R_{3} - (- 14 + \sqrt{21}) R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{3 - \sqrt{21}}{3} & - \frac{3 - \sqrt{21}}{6} & 0 \\ 0 & 1 & - \frac{7 + \sqrt{21}}{14} & 0 \\ 0 - (- 14 + \sqrt{21}) \cdot 0 & - 14 + \sqrt{21} - (- 14 + \sqrt{21}) \cdot 1 & \frac{11 + \sqrt{21}}{2} - (- 14 + \sqrt{21}) (- \frac{7 + \sqrt{21}}{14}) & 0 - (- 14 + \sqrt{21}) \cdot 0 \end{array}]$

Step 5.3.2.5.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & \frac{3 - \sqrt{21}}{3} & - \frac{3 - \sqrt{21}}{6} & 0 \\ 0 & 1 & - \frac{7 + \sqrt{21}}{14} & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Step 5.3.2.6

Perform the row operation $R_{1} = R_{1} - \frac{3 - \sqrt{21}}{3} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Step 5.3.2.6.1

Perform the row operation $R_{1} = R_{1} - \frac{3 - \sqrt{21}}{3} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - \frac{3 - \sqrt{21}}{3} \cdot 0 & \frac{3 - \sqrt{21}}{3} - \frac{3 - \sqrt{21}}{3} \cdot 1 & - \frac{3 - \sqrt{21}}{6} - \frac{3 - \sqrt{21}}{3} (- \frac{7 + \sqrt{21}}{14}) & 0 - \frac{3 - \sqrt{21}}{3} \cdot 0 \\ 0 & 1 & - \frac{7 + \sqrt{21}}{14} & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Step 5.3.2.6.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & - \frac{7 - \sqrt{21}}{14} & 0 \\ 0 & 1 & - \frac{7 + \sqrt{21}}{14} & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Step 5.3.3

Use the result matrix to declare the final solution to the system of equations.

$x - \frac{7 - \sqrt{21}}{14} z = 0$

$y - \frac{7 + \sqrt{21}}{14} z = 0$

$0 = 0$

Step 5.3.4

Write a solution vector by solving in terms of the free variables in each row.

$[\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} \frac{z}{2} - \frac{z \sqrt{21}}{14} \\ \frac{z}{2} + \frac{z \sqrt{21}}{14} \\ z \end{array}]$

Step 5.3.5

Write the solution as a linear combination of vectors.

$[\begin{array}{c} x \\ y \\ z \end{array}] = z [\begin{array}{c} \frac{1}{2} - \frac{\sqrt{21}}{14} \\ \frac{1}{2} + \frac{\sqrt{21}}{14} \\ 1 \end{array}]$

Step 5.3.6

Write as a solution set.

${z [\begin{array}{c} \frac{1}{2} - \frac{\sqrt{21}}{14} \\ \frac{1}{2} + \frac{\sqrt{21}}{14} \\ 1 \end{array}] | z \in R}$

Step 5.3.7

The solution is the set of vectors created from the free variables of the system.

${[\begin{array}{c} \frac{1}{2} - \frac{\sqrt{21}}{14} \\ \frac{1}{2} + \frac{\sqrt{21}}{14} \\ 1 \end{array}]}$

Step 6

The eigenspace of $A$ is the list of the vector space for each eigenvalue.

${[\begin{array}{c} 14 \\ 4 \\ 1 \end{array}], [\begin{array}{c} \frac{1}{2} + \frac{\sqrt{21}}{14} \\ \frac{1}{2} - \frac{\sqrt{21}}{14} \\ 1 \end{array}], [\begin{array}{c} \frac{1}{2} - \frac{\sqrt{21}}{14} \\ \frac{1}{2} + \frac{\sqrt{21}}{14} \\ 1 \end{array}]}$